Boundary Value Problems

1 Fourier Series and Integrals 1 Periodic Functions and Fourier Series 2 Arbitrary Period. Periodic Extensions 3 Even and Odd Functions. Half-Range Expansions 4 Convergence of Fourier Series 5 Uniform Convergence 6 Summability of Fourier Series 7 Mean Error and Convergence in Mean 8 Numerical Determination of Fourier Coefficients 9 Complex Methods 10 Fourier Integral 11 Applications of Fourier Series and Integrals 12 Comments and References2 The Heat Equation 1 Derivation and Boundary Conditions 2 Example: Fixed End Temperatures 3 Example: Insulated Bar 4 Example: Convection 5 Sturm-Liouville Problems 6 Expansion in Series of Eigenfunctions 7 Generalities on the Heat Conduction Problem 8 Semi-Infinite Rod 9 Infinite Rod 10 Comments and References3 The Wave Equation 1 The Vibrating String 2 Solution of the Vibrating String Problem 3 D'Alembert's Solution 4 Generalities on the One-Dimensional Wave Equation 5 Estimation of Eigenvalues 6 Wave Equation in Unbounded Regions 7 References4 The Potential Equation 1 Potential Equation 2 Potential in a Rectangle 3 Potential in a Slot 4 Potential in a Disk 5 Classification of Partial Differential Equations and Limitations of the Product Method 6 Comments and References5 Problems in Several Dimensions 1 Derivation of the Two-Dimensional Wave Equation 2 Derivation of the Two-Dimensional Heat Equation 3 Solution of the Two-Dimensional Heat Equation 4 Problems in Polar Coordinates 5 Bessel's Equation 6 Vibrations of a Circular Membrane 7 Some Applications of Bessel Functions 8 Spherical Coordinates. Legendre Polynomials 9 References6 Laplace Transform 1 Definition and Elementary Properties 2 Elementary Applications. Partial Fractions and Convolutions 3 Applications to Partial Differential Equations 4 More Difficult Examples 5 Comments and References7 Numerical Methods 1 Boundary Value Problems 2 The Heat Equation 3 Two-Dimensional Heat Equation 4 Potential Equation 5 ReferencesBibliographyAnswers to Selected ExercisesIndex